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www.nature.com/scientificreports OPEN Quasiparticle Properties under Interactions in Weyl and Nodal Line Semimetals Received: 23 July 2018 Jing Kang 1, Jianfei Zou1, Kai Li2, Shun-Li Yu3,4 & Lu-Bing Shao3,4 Accepted: 22 January 2019 The quasiparticle spectra of interacting Weyl and nodal-line semimetals on a cubic lattice are studied Published: xx xx xxxx using the cluster perturbation theory. By tracking the spectral functions under interaction, we fnd that the Weyl points will move to and meet at a specifc point in one Weyl semimetal model, while in the other Weyl semimetal model they are immobile. In the nodal-line semimetals, we fnd that the nodal line shrinks to a point and then disappears under interaction in one-nodal-line system. When we add another nodal line to this system, we fnd that the two nodal lines both shrink to specifc points, but the disappearing processes of the two nodal lines are not synchronized. We argue that the nontrivial evolution of Weyl points and nodal lines under interaction is due to the presence of symmetry breaking order, e.g., a ferromagnetic moment, in the framework of mean feld theory, whereas the stability of Weyl points under interaction is protected by symmetry. Among all these models, the spectral gap is fnally opened when the interaction is strong enough. Te discovery of semimetallic features in electronic band structures protected by the interplay of symmetry and topology provides a realistic platform for the concepts of fundamental physics theory in condensed matter phys- ics. In recent years, Weyl semimetals1–4 (WSM) have been attracted considerable attention since they extend the topological classifcation of mater beyond the insulators and exhibit the exotic Fermi arc surface state5. WSM 6,7 8,9 materials such as the TaAs-family pnictides and MoTe2 have been discovered by observation of the unique Fermi arcs of surface states through angle-resolved photoemission spectroscopy10–15. In these materials, the con- duction and valence bands touch each other around several points, which are called Weyl points (WPs), in the momentum space. Te WP acts as a topological monopole which can be quantifed by corresponding chiral charge through calculating the fux of Berry curvature16. According to the conservation of chirality, the WPs always exist in pairs of opposite chirality. Te topological Fermi arcs are arising from the connection of two projections of the bulk WPs with two opposite chiral charges in the surface Brillouin zone (BZ). WSMs and their surface states may lead to unusual spectroscopic and transport phenomena such as chiral anomaly, spin and anomalous Hall efects9. Te WSMs have a Fermi surface consisting of a fnite number of WPs in the BZ, at which the conduction and valence bands meet linearly. Te WPs are twofold degeneracy and only exist in condensed matter systems with breaking either time-reversal or spatial-inversion symmetry. Such a phase has massless Weyl quasiparticles which can be viewed as half-Dirac Fermions. Besides, the Weyl quasiparticles can be gapped by coupling two quasiparticles with diferent chirality17. In the presence of interactions which can easily arise in realistic systems, the Weyl quasiparticles can be moved, normalized and even gapped. Another interesting system is the nodal-line semimetal(NLSM) with one-dimensional Fermi surfaces18–21. In the experiment side, the NLSMs were observed 22 23,24 in several compounds such as PbTaSe2 and ZrSiS . In this system there are bulk band touchings along 1D lines and these line-like touchings need extra symmetries to be topologically protected. Interactions can also be applied in this system to discuss the proximity efect and spontaneous symmetry breaking25. In this paper, by using the Cluster Perturbation Teory (CPT)26–31, we study two lattice models for WSMs and one for NLSM to see how the on-site Coulomb interaction afects the WPs and nodal lines. In CPT, the quasipar- ticle spectral function can be calculated and then the positions of the WPs and nodal lines can be tracked through 1College of Science, Hohai University, Nanjing, 210098, China. 2School of Physics and Engineering, Zhengzhou University, Zhengzhou, 450001, China. 3National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, 210093, China. 4Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, 210093, China. Correspondence and requests for materials should be addressed to J.K. (email: [email protected]) or K.L. (email: [email protected]) SCIENTIFIC REPORTS | (2019) 9:2824 | https://doi.org/10.1038/s41598-019-39258-3 1 www.nature.com/scientificreports/ www.nature.com/scientificreports Figure 1. (a) A 2 × 2 × 2-site cluster, as marked by numbers 1–8, in the cubic lattice. (b) Te scanning routine is Γ-X-M-Γ-R-X′-Γ in the bulk BZ. the spectral function when the interaction alters. We fnd that the WPs will move to and meet each other under interactions in one WSM model, while in the other WSM model they will not. In the one-nodal-line semimetal system, the nodal line shrinks to a point under interactions. When we add another nodal line to this system, we fnd that the two nodal lines will all shrink to specifc points one afer the other. We argue that the nontrivial evolution of WPs and nodal lines under interaction is due to the presence of symmetry breaking order, e.g., a ferromagnetic moment, in the framework of mean feld theory, whereas the stability of WPs under interaction is protected by symmetry. Among all these models, the spectral gap is fnally opened when the interaction is strong enough. Models and Methods Weyl Semimetal Models. We will consider two kinds of Weyl Semimetal Models. For the frst kind of WSM model (WSM1), the tight-binding Hamiltonian is written as follows16,32: =−† +− − σ Hc00∑k kx{[2(tkcoscoskm)(2coskkyzcos)] x ++2stkin yyσσ2stkin zz},ck (1) ††† where ckk= (,cc↑↓k ) and σx,y,z are the Pauli matrices. Te hopping constant t will be set t = 1 in calculations based on this model. Tis model breaks both time-reversal and space-inversion symmetries. Afer diagonalizing this non-interacting Hamiltonian, we can get the band structure =± −+−− 22++2 . εkx[2(coskkcos)0 mk(2 coscyzoskk)] 4sin yz4sin k (2) Assuming εk = 0, we get the WPs in the bulk BZ. Tere can be 2, 6 or 8 WPs for diferent parameter m, while two of the WPs are present at (±k0, 0, 0). We now introduce the second kind of WSM model (WSM2) which, in contrast to both the WSM1 and the second Weyl semimetal model studied in ref.16, preserves the space-inversion symmetry but not the time-reversal symmetry. Te tight-binding Hamiltonian is written as follows: =+† σσ+.σ Ht0 2[∑kckxcos(kkk)cxyos()yzcos( )]zkc (3) Similar to the second Weyl semimetal model studied in ref.16, the real space hopping is a type of bond-selective and spin-dependent Kitaev-like hopping31. Afer diagonalizing this non-interacting Hamiltonian, we can get the band structure =± 222++. εkx2ct os kkkcoscyzos (4) πππ Tis model gives us eight WPs at ±±,,± . ( 222) Nodal-Line Semimetal Model. For the NLSM model, the tight-binding Hamiltonian is written as follows25: † Hc01=+∑ kx{2[(tkcoscoskbyz−+)(tk23cos1−+)]σσxz2stkin yk},c k (5) SCIENTIFIC REPORTS | (2019) 9:2824 | https://doi.org/10.1038/s41598-019-39258-3 2 www.nature.com/scientificreports/ www.nature.com/scientificreports Figure 2. Intensity map of the spectral function A(k, ω) along high symmetry lines (see Fig. 1(b)) with m = 1.5 and k0 = 0.2π based on the WSM1 model for (a) U = 0, (b) U = 3, (c) U = 8 and (d) U = 9, respectively. Te red dashed lines denote the non-interacting band structures. where t1, t2, t3 are the hopping constants. We can also diagonalize the non-interacting Hamiltonian to get the band structure, which reads: =± +−+−2 +.22 εkx2[tk12(cos cos)kbyztk(cos 1)]stk3 in z (6) During our calculations based on this model, the hopping constant t1 will be set as the energy unit t1 = 1. By setting proper parameters t2, b, we can get one or two nodal lines. Interaction Hamiltonian. In order to investigate the efects of interactions in the semimetals, we use the Hubbard model =+ − μ HH0,Un∑∑i ii↑↓nn,,i i σ, (7) where H0 denotes a tight-binding Hamiltonian of the non-interacting semimetals as we have introduced above. U is the on-site Coulomb interaction, ni,σ is the particle-number operator on site i with spin σ and μ is the chemical potential which will be fxed μ = 1 U at half-flling as the band structures hold particle-hole symmetry. 2 Numerical Method. A good way to analyze the Hubbard model numerically is the exact diagonalization (ED). However, due to the limitation of computer memory capacity and speed of CPU, the solvable size of the lat- tice cannot be large. Tus one will not get enough momentum points to study the k-space distribution of the spec- tral function. Te CPT which is based on the ED is another numerical method to solve the Hubbard model26–28. Te Green’s function in a fnite size of lattice is frst calculated through the ED. Ten we can divide the whole lattice into many small lattices of this size which are usually called clusters. Te inter-cluster hopping is treated perturbatively and the Green’s function of the physical system is obtained from strong-coupling perturbation theory. As we get enough momentum points, we can track the evolution of WPs or nodal lines controlled by the variation of U. During our calculations, we choose a 2 × 2 × 2-site cluster in the cubic lattice shown in Fig.
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